Field
The present systems, methods and apparatus generally relate to quantum computation and specifically relate to superconducting quantum computation and implementations of quantum annealing.
Superconducting Qubits
There are many different hardware and software approaches under consideration for use in quantum computers. One hardware approach employs integrated circuits formed of superconducting material, such as aluminum and/or niobium, to define superconducting qubits. Superconducting qubits can be separated into several categories depending on the physical property used to encode information. For example, they may be separated into charge, flux and phase devices. Charge devices store and manipulate information in the charge states of the device; flux devices store and manipulate information in a variable related to the magnetic flux through some part of the device; and phase devices store and manipulate information in a variable related to the difference in superconducting phase between two regions of the phase device.
Many different forms of superconducting flux qubits have been implemented in the art, but all successful implementations generally include a superconducting loop (i.e., a “qubit loop”) that is interrupted by at least one Josephson junction. Some embodiments implement multiple Josephson junctions connected either in series or in parallel (i.e., a compound Josephson junction) and some embodiments implement multiple superconducting loops.
Persistent Current
As previously discussed, a superconducting flux qubit may comprise a qubit loop that is interrupted by at least one Josephson junction, or at least one compound Josephson junction. Since a qubit loop is superconducting, it effectively has no electrical resistance. Thus, electrical current traveling in a qubit loop may experience no dissipation. If an electrical current is induced in the qubit loop by, for example, a magnetic flux signal, this current may be sustained indefinitely. The current may persist indefinitely until it is interfered with in some way or until the qubit loop is no longer superconducting (due to, for example, heating the qubit loop above its critical temperature). For the purposes of this specification, the term “persistent current” is used to describe an electrical current circulating in a qubit loop of a superconducting qubit. The sign and magnitude of a persistent current may be influenced by a variety of factors, including but not limited to a flux signal φX coupled directly into the qubit loop and a flux signal φCJJ coupled into a compound Josephson junction that interrupts the qubit loop.
Quantum Processor
A computer processor may take the form of an analog processor, for instance a quantum processor such as a superconducting quantum processor. A superconducting quantum processor may include a number of qubits and associated local bias devices, for instance two or more superconducting qubits. Further detail and embodiments of exemplary quantum processors that may be used in conjunction with the present systems, methods, and apparatus are described in US Patent Publication No. 2006-0225165, US Patent Publication 2008-0176750, U.S. patent application Ser. No. 12/266,378, and PCT Patent Application Serial No. PCT/US09/37984.
Adiabatic Quantum Computation
Adiabatic quantum computation typically involves evolving a system from a known initial Hamiltonian (the Hamiltonian being an operator whose eigenvalues are the allowed energies of the system) to a final Hamiltonian by gradually changing the Hamiltonian. A simple example of an adiabatic evolution is:He=(1−s)Hi+sHf 
where Hi is the initial Hamiltonian, Hf is the final Hamiltonian, He is the evolution or instantaneous Hamiltonian, and s is an evolution coefficient which controls the rate of evolution. As the system evolves, the coefficient s goes from 0 to 1 such that at the beginning (i.e., s=0) the evolution Hamiltonian He is equal to the initial Hamiltonian Hi and at the end (i.e., s=1) the evolution Hamiltonian He is equal to the final Hamiltonian Hf. Before the evolution begins, the system is typically initialized in a ground state of the initial Hamiltonian Hi and the goal is to evolve the system in such a way that the system ends up in a ground state of the final Hamiltonian Hf at the end of the evolution. If the evolution is too fast, then the system can be excited to a higher energy state, such as the first excited state. In the present systems, methods, and apparatus, an “adiabatic” evolution is considered to be an evolution that satisfies the adiabatic condition:{dot over (s)}|1|dHe/ds|0|=δg2(s)where {dot over (s)} is the time derivative of s, g(s) is the difference in energy between the ground state and first excited state of the system (also referred to herein as the “gap size”) as a function of s, and δ is a coefficient much less than 1.
The evolution process in adiabatic quantum computing may sometimes be referred to as annealing. The rate that s changes, sometimes referred to as an evolution or annealing schedule, is normally slow enough that the system is always in the instantaneous ground state of the evolution Hamiltonian during the evolution, and transitions at anti-crossings (i.e., when the gap size is smallest) are avoided. Further details on adiabatic quantum computing systems, methods, and apparatus are described in U.S. Pat. No. 7,135,701.
Quantum Annealing
Quantum annealing is a computation method that may be used to find a low-energy state, typically preferably the ground state, of a system. Similar in concept to classical annealing, the method relies on the underlying principle that natural systems tend towards lower energy states because lower energy states are more stable. However, while classical annealing uses classical thermal fluctuations to guide a system to its global energy minimum, quantum annealing may use quantum effects, such as quantum tunneling, to reach a global energy minimum more accurately and/or more quickly. It is known that the solution to a hard problem, such as a combinatorial optimization problem, may be encoded in the ground state of a system Hamiltonian and therefore quantum annealing may be used to find the solution to such hard problems. Adiabatic quantum computation is a special case of quantum annealing for which the system, ideally, begins and remains in its ground state throughout an adiabatic evolution. Thus, those of skill in the art will appreciate that quantum annealing systems and methods may generally be implemented on an adiabatic quantum computer, and vice versa. Throughout this specification and the appended claims, any reference to quantum annealing is intended to encompass adiabatic quantum computation unless the context requires otherwise.
Quantum annealing is an algorithm that uses quantum mechanics as a source of disorder during the annealing process. The optimization problem is encoded in a Hamiltonian HP, and the algorithm introduces strong quantum fluctuations by adding a disordering Hamiltonian HD that does not commute with HP. An example case is:HE=HP+ΓHD,
where Γ changes from a large value to substantially zero during the evolution and HE may be thought of as an evolution Hamiltonian similar to He described in the context of adiabatic quantum computation above. The disorder is slowly removed by removing HD (i.e., reducing Γ). Thus, quantum annealing is similar to adiabatic quantum computation in that the system starts with an initial Hamiltonian and evolves through an evolution Hamiltonian to a final “problem” Hamiltonian HP whose ground state encodes a solution to the problem. If the evolution is slow enough, the system will typically settle in a local minimum close to the exact solution; the slower the evolution, the better the solution that will be achieved. The performance of the computation may be assessed via the residual energy (distance from exact solution using the objective function) versus evolution time. The computation time is the time required to generate a residual energy below some acceptable threshold value. In quantum annealing, HP may encode an optimization problem and therefore HP may be diagonal in the subspace of the qubits that encode the solution, but the system does not necessarily stay in the ground state at all times. The energy landscape of HP may be crafted so that its global minimum is the answer to the problem to be solved, and low-lying local minima are good approximations.
The gradual reduction of Γ in quantum annealing may follow a defined schedule known as an annealing schedule. Unlike traditional forms of adiabatic quantum computation where the system begins and remains in its ground state throughout the evolution, in quantum annealing the system may not remain in its ground state throughout the entire annealing schedule. As such, quantum annealing may be implemented as a heuristic technique, where low-energy states with energy near that of the ground state may provide approximate solutions to the problem.
Fixed Quantum Annealing with a Superconducting Quantum Processor
A straightforward approach to quantum annealing with superconducting flux qubits uses fixed flux biases applied to the qubit loops (φX) and qubit couplers (φJ). The motivation of this scheme is to define the problem Hamiltonian HP by these fixed flux biases, which generally remain static throughout the annealing process. The disorder term ΓHD may be realized by, for example, coupling a respective flux signal φCJJ into the compound Josephson junction of each ith qubit to realize single qubit tunnel splitting Δi. In the annealing procedure, the φCJJ signals are initially applied to induce maximum disorder in each qubit and then gradually varied such that only HP, as defined by the static flux biases, remains at the end of the evolution. This approach, referred to herein as “fixed quantum annealing” because the signals applied to the qubit loops remain substantially fixed, is attractive due to its simplicity: the only time varying signals are applied to the qubit compound Josephson junctions in order to modulate the tunnel splitting Δ. However, this approach does not account for an important effect: qubit persistent currents are also a function of the flux signal φCJJ applied to the compound Josephson junction of each qubit. This means that the carefully crafted terms of the problem Hamiltonian HP that are intended to be defined by the static flux biases applied to the qubit loops (φX) and qubit couplers (φJ) are actually influenced by the gradual reduction of the φCJJ signals in the annealing process. Simply applying fixed flux biases (φX and φJ) does not address this issue. The fact that the qubit persistent currents evolve during the annealing process may affect the whole evolution path of the system.
The ultimate goal of quantum annealing is to find a low-energy state, typically preferably the ground state, of a system Hamiltonian. The specific system Hamiltonian for which the low-energy state is sought is the problem Hamiltonian HP which is characterized, at least in part, by the persistent currents circulating in each respective qubit. In quantum annealing the problem Hamiltonian HP is typically configured right from the beginning. The annealing procedure then involves applying a disorder term ΓHD (which realizes the tunnel splitting Δ) that effectively smears the state of the system, and then gradually removing this disorder term such that the system ultimately stabilizes in a low-energy state (such as the ground state) of the problem Hamiltonian HP. In the fixed quantum annealing approach, the terms of HP are statically applied throughout the annealing process and the only time-varying signals are the φCJJ signals that realize the disorder term ΓHD. However, because the qubit persistent currents are ultimately influenced by the application and gradual removal of the φCJJ signals, the energy landscape of the problem Hamiltonian HP varies throughout the annealing procedure. This means that while the annealing procedure seeks a low-energy state of HP, the problem Hamiltonian HP itself evolves and so too does the location of the desired low-energy state (e.g., ground state). Furthermore, the “gradual removal” of the disorder term ΓHD is typically physically achieved by a series of downward steps as opposed to a continuous ramping. Because the persistent current in the qubits changes in response to each downward step, the system may effectively anneal towards a different state at each step. Thus, fixed quantum annealing with superconducting flux qubits can be problematic because it relies on a discontinuous evolution towards a moving target. As such, there is a need in the art for a more reliable and accurate protocol for quantum annealing with superconducting flux qubits.